Forecasting |
The optimal (minimum MSE) -step-ahead forecast of
is
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with and
for
. For the forecasts
, see the section State-Space Representation.
Under the stationarity assumption, the optimal (minimum MSE) -step-ahead forecast of
has an infinite moving-average form,
. The prediction error of the optimal
-step-ahead forecast is
, with zero mean and covariance matrix,
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where with a lower triangular matrix
such that
. Under the assumption of normality of the
, the
-step-ahead prediction error
is also normally distributed as multivariate
. Hence, it follows that the diagonal elements
of
can be used, together with the point forecasts
, to construct
-step-ahead prediction intervals of the future values of the component series,
.
The following statements use the COVPE option to compute the covariance matrices of the prediction errors for a VAR(1) model. The parts of the VARMAX procedure output are shown in Figure 35.36 and Figure 35.37.
proc varmax data=simul1; model y1 y2 / p=1 noint lagmax=5 printform=both print=(decompose(5) impulse=(all) covpe(5)); run;
Figure 35.36 is the output in a matrix format associated with the COVPE option for the prediction error covariance matrices.
Prediction Error Covariances | |||
---|---|---|---|
Lead | Variable | y1 | y2 |
1 | y1 | 1.28875 | 0.39751 |
y2 | 0.39751 | 1.41839 | |
2 | y1 | 2.92119 | 1.00189 |
y2 | 1.00189 | 2.18051 | |
3 | y1 | 4.59984 | 1.98771 |
y2 | 1.98771 | 3.03498 | |
4 | y1 | 5.91299 | 3.04856 |
y2 | 3.04856 | 4.07738 | |
5 | y1 | 6.69463 | 3.85346 |
y2 | 3.85346 | 5.07010 |
Figure 35.37 is the output in a univariate format associated with the COVPE option for the prediction error covariances. This printing format more easily explains the prediction error covariances of each variable.
Prediction Error Covariances by Variable | |||
---|---|---|---|
Variable | Lead | y1 | y2 |
y1 | 1 | 1.28875 | 0.39751 |
2 | 2.92119 | 1.00189 | |
3 | 4.59984 | 1.98771 | |
4 | 5.91299 | 3.04856 | |
5 | 6.69463 | 3.85346 | |
y2 | 1 | 0.39751 | 1.41839 |
2 | 1.00189 | 2.18051 | |
3 | 1.98771 | 3.03498 | |
4 | 3.04856 | 4.07738 | |
5 | 3.85346 | 5.07010 |
Exogenous variables can be both stochastic and nonstochastic (deterministic) variables. Considering the forecasts in the VARMAX(,
,
) model, there are two cases.
As defined in the section State-Space Representation, has the representation
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and hence
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Therefore, the covariance matrix of the -step-ahead prediction error is given as
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where is the covariance of the white noise series
, and
is the white noise series for the VARMA(
,
) model of exogenous (independent) variables, which is assumed not to be correlated with
or its lags.
The optimal forecast of
conditioned on the past information and also on known future values
can be represented as
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and the forecast error is
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Thus, the covariance matrix of the -step-ahead prediction error is given as
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In the relation , the diagonal elements can be interpreted as providing a decomposition of the
-step-ahead prediction error covariance
for each component series
into contributions from the components of the standardized innovations
.
If you denote the ()th element of
by
, the MSE of
is
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Note that is interpreted as the contribution of innovations in variable
to the prediction error covariance of the
-step-ahead forecast of variable
.
The proportion, , of the
-step-ahead forecast error covariance of variable
accounting for the innovations in variable
is
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The following statements use the DECOMPOSE option to compute the decomposition of prediction error covariances and their proportions for a VAR(1) model:
proc varmax data=simul1; model y1 y2 / p=1 noint print=(decompose(15)) printform=univariate; run;
The proportions of decomposition of prediction error covariances of two variables are given in Figure 35.38. The output explains that about 91.356% of the one-step-ahead prediction error covariances of the variable is accounted for by its own innovations and about 8.644% is accounted for by
innovations.
Proportions of Prediction Error Covariances by Variable |
|||
---|---|---|---|
Variable | Lead | y1 | y2 |
y1 | 1 | 1.00000 | 0.00000 |
2 | 0.88436 | 0.11564 | |
3 | 0.75132 | 0.24868 | |
4 | 0.64897 | 0.35103 | |
5 | 0.58460 | 0.41540 | |
y2 | 1 | 0.08644 | 0.91356 |
2 | 0.31767 | 0.68233 | |
3 | 0.50247 | 0.49753 | |
4 | 0.55607 | 0.44393 | |
5 | 0.53549 | 0.46451 |
If the CENTER option is specified, the sample mean vector is added to the forecast.
If dependent (endogenous) variables are differenced, the final forecasts and their prediction error covariances are produced by integrating those of the differenced series. However, if the PRIOR option is specified, the forecasts and their prediction error variances of the differenced series are produced.
Let be the original series with some appended zero values that correspond to the unobserved past observations. Let
be the
matrix polynomial in the backshift operator that corresponds to the differencing specified by the MODEL statement. The off-diagonal elements of
are zero, and the diagonal elements can be different. Then
.
This gives the relationship
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where and
.
The -step-ahead prediction of
is
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The -step-ahead prediction error of
is
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Letting , the covariance matrix of the l-step-ahead prediction error of
,
, is
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If there are stochastic exogenous (independent) variables, the covariance matrix of the l-step-ahead prediction error of ,
, is
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